Electroweak symmetry breaking in supersymmetric models with heavy scalar superpartners

We propose a novel mechanism of electroweak symmetry breaking in supersymmetric models, as the one recently discussed by Birkedal, Chacko and Gaillard, in which the Standard Model Higgs doublet is a pseudo-Goldstone boson of some global symmetry. The Higgs mass parameter is generated at one loop level by two different, moderately fine-tuned sources of the global symmetry breaking. The mechanism works for scalar superpartner masses of order 10 TeV, but gauginos can be light. The scale at which supersymmetry breaking is mediated to the visible sector has to be low, of order 100 TeV. Fine-tuning in the scalar potential is at least two orders of magnitude smaller than in the MSSM with similar soft scalar masses. The physical Higgs boson mass is (for $\tan\beta\gg1$) in the range 120-135 GeV.


Introduction
Explanation of the origin of the Fermi scale is a challenge for physics beyond the Standard Model (SM) 5 An interesting possibility is that the electroweak symmetry breaking is generated by quantum corrections to the Higgs doublet potential. For such a mechanism to be under theoretical control, the tree level Higgs mass squared parameter m 2 H has to be calculable (at least in principle) at some scale Λ S in terms of some more fundamental parameters. Secondly, the dependence of quantum corrections to the Higgs doublet potential on dimensionful parameters of new physics should be moderate. Otherwise large cancellations between the tree level parameters and quantum corrections would be necessary, rendering such a mechanism doubtful. Although fine-tuning is difficult to quantify in a precise way, it is usually easy to give a rough estimate of its order of magnitude. As a reference number it is worth remembering that in the SM cut-off by the neutrino see-saw scale of order 10 13 GeV, the Higgs potential has to be fine-tuned in 1 part to 10 20 .
Radiative electroweak symmetry breaking, triggered by the large top quark Yukawa coupling, has been a succesful prediction of the Minimal Supersymmetric Standard Model (MSSM). However, in the MSSM the necessary degree of fine-tuning in the Higgs potential grows exponentially with the value of the lightest CP-even Higgs boson mass [3]. The present LEP limits on the Higgs boson mass push the stop masses into the range 500 GeV−1 TeV and, in consequence, the necessary fine-tuning in the MSSM Higgs potential is estimated to be of order of 1% [4,5]. This fact may be taken as somewhat dissapointing for a supersymmetric model and it stimulated several authors to look for alternatives to the MSSM [6] and to supersymmetry itself, which could explain the origin of the Fermi scale [7,8,9,10]. However, no convincing idea has emerged yet that would lead to fine-tuning significantly lower than O(1%) needed in the MSSM with the present limits on the Higgs boson mass.
We believe, therefore, it may be worthwhile to ask a different question. After all, the questions about an acceptable degree of fine-tuning and even about its definition do not have any sensible quantitative answer at the level of effective theories, for instance, when the theory of soft supersymmetry breaking terms is not known [4,11]. More bothersome is the quadratic dependence of the fine-tuning in the MSSM on the mass of the scalar su-perparners of the top quark. Although the FCNC effects in the MSSM are controlled by the squark masses of the first two families, a light stop is not easy (although not impossible) to reconcile with the observed suppression of the FCNC effects [12]. Indeed, unless the first two sfermion families are degenerate in mass, they have to be heavy, with masses at least O(10 TeV), and the large splitting with the third one needs some explanation by the mechanism of supersymmetry breaking [13]. Thus it is of some interest to ask if in models in which radiative electroweak symmetry breaking occurs one can avoid quadratic dependence of the Fermi scale on the stop masses, i.e. if one can significantly rise the scalar superpartner masses without jeopardizing naturalness.
An interesting and very simple combination of the idea of the Higgs doublet as a pseudo-Goldstone boson (revived in the non-supersymmetric Little Higgs models [8], inspired by the so-called deconstruction [14]) and of supersymmetry has been proposed recently by Birkedal, Chacko and Gaillard [15]. In their model the Higgs doublet is a Goldstone boson of a spontaneously broken global SU(3) symmetry. Global SU(3) is also explicitly broken by a supersymmetric fermion mass term and by the SM SU(2) L × U(1) Y electroweak interactions. In the present paper we show that in this model, in certain range of its parameters, an interesting mechanism of radiative electroweak symmetry breaking can be realized. The Higgs doublet mass parameter is then generated at one-loop level by the two moderately fine-tuned sources of the global SU(3) symmetry breaking. The mechanism works for scalar superpartner masses of order O(10 TeV). Fine-tuning in the scalar potential is at least two orders of magnitude smaller than in the MSSM with similar soft scalar masses, i.e. stays at the level of (1%). The physical Higgs boson mass is predicted to be (for tan β ≫ 1) in the range 120 − 135 GeV, where the main source of uncertainty are unknown two-loop effects.

The model
In this section we introduce our model which is a slight modificaton of the one proposed in [15]. The Higgs and the 3rd generation weak doublet superfields are extended to fit the fundamental (Ĥ u ) and anti-fundamental (Ĥ d andQ) representations of an approximate global SU(3) symmetry: In addition there is a new SU(3) singlet quark supermultipletT c . At some high scale Λ S the SU(3) symmetry is respected by the top Yukawa couplings in the superpotential, but is explicitly broken by the µ T term: In order to preserve the symmetry between the up-and down-type sectors one can introduce another global SU(3) ′ symmetry, that controls the bottom sector. To this end we extend the quark doubletQ 3 by the SU(2) L singlet quark superfieldB, so thatQ is in the fundamental representation of SU(3) ′ (with respect to whichĤ u andĤ d form the fundamental and anti-fundamental representations, respectively). We also introduce a corresponding SU(3) ′ singlet quark superfield B c . The bottom sector superpotential then reads Here SU(3) ′ is explicitly broken by µ B . Of course, w b breaks SU(3) while w t breaks SU(3) ′ . We also assume that at the high scale Λ S the SU(3) and SU(3) ′ symmetries are respected by the soft mass terms and trilinear couplings in the top and bottom sectors, respectively. Thus, the most general form of these mass terms (at Λ S ) is: 6 Finally, the Higgs doublet µ term is assumed to respect the SU(3) (and in fact, also the SU(3) ′ ) symmetry As in ref. [15] the gauge symetry of the MSSM is extended by the U(1) E group, which commutes with the global SU(3) and SU(3) ′ symmetries. The where m 2 can be achieved in a similar way as the electroweak symmetry breaking in the MSSM, i.e. by quantum corrections induced by the large Yukawa coupling Y T in the superpotential (2) [15]. Similarly as in the MSSM, the mass of the additional neutral Z ′ boson is then given by: There is also the tree-level SU(3) breaking potential originating from the electroweak D-terms: Let us now identify the SM Higgs doublet. The SU(3) global symmetry is spontaneously broken down to SU(2) by the vevs (8) which leads to where |H| ≡ |H † H|. In the following we will keep track only of the real neutral component of the Higgs doublet H, i.e. we will substitute (0, h) T for H and h for |H|. With this parametrization it is explicit that the SU (3) preserving Higgs potential (7) does not contribute to the potential of the SM Higgs doublet H. At the tree level the SM Higgs potential has only the quartic part which arises from the electroweak D-terms (10) explicitly breaking global SU(3):

One loop SM Higgs potential
Since the SU(3) symmetry is only approximate, corrections to the SM Higgs potential appear at loop level. We therefore calculate the one-loop effective potential V = V tree + ∆V 1−loop in terms of the Lagrangian parameters renormalized at the scale Λ S . In a supersymmetric model such a calculation may be equivalently viewed as the calculation in terms of the bare parameters and with the momentum cut-off Λ S . In a consistent one-loop calculation of the effective potential the SU(3) symmetric parametres defined by eq. (5), (7), (6), (2) and (4), must be used. An SU(3) splitting of these parameters is generated at one loop level, too, by the two sources of explicit SU (3) breaking: the nonzero electroweak gauge couplings g 2 , g y and the µ T and µ B terms. It enters the effective potential only in the two-loop approximation. For the mechanism of the electroweak symmetry breaking we propose in this paper such higher order effects must be negligible and this constrains the scale Λ S at which the soft mass terms are generated. As we shall see, for squarks masses O(10 TeV) our mechanism works anyway only for Λ S ∼ O(100 TeV), consistently with the above requirement. Moreover, we assume that the SU (3) breaking corrections to the Yukawa couplings also vanish above Λ S , i.e. that above Λ S the model described by eqs. (2), (4), (5), (6) and (7) is embedded in some SU(3) invariant theory as in [15]. Before presenting our results for the one-loop effective potential we recall the structure of the effective potential in other models. In the MSSM quadratically divergent corrections to the Higgs mass parameter are absent at any order of perturbation theory due to supersymmetry. Logarithmically divergent contribution is determined by STrM 4 and depends quadratically on the supersymmetry breaking mass parameters. It consists of two parts: one proportional to the top Yukawa coupling quadratically dependent on the sfermion and Higgs fields soft mass parameters and one proportional to the gauge couplings and quadratically dependent on gaugino masses. In the nonsupersymmetric Little Higgs Models [8] global symmetries forbid quadratically divergent corrections to the Higgs mass parameter at one-loop (but such divergences are present already at two-loops). In the language of the effective potential, the mass matrix squared M 2 in these models does not depend on the SM Higgs field h and, what is important, the cancellations occur independently in fermionic and bosonic sectors. Logarithmically divergent corrections proportional to the top Yukawa coupling are generically present already at the one-loop level and they, as well as the two-loop quadrati-cally divergent ones, require some ultraviolet completion of the models at the scales of order 10 TeV. Finally, there are models with "hard" supersymmetry breaking but such, that the leading contribution to the effective potential proportional to the top quark Yukawa coupling is finite at one-loop [7]. However quadratically divergent contributions appear from D-terms [19] and at higher orders and require a low cut-off.
In the model discussed in this paper the situation is still different and more elegant, as long as all SU(3) breaking quantum effects in the parameters entering ∆V 1−loop can be neglected (i.e. for a low scale Λ S ). Because the model is supersymmetric, all quadratic divergences are absent to all orders in perturbation theory. Moreover, the SM Higgs potential must be proportional to some parameter that breaks SU(3) symmetry and also leads to supersymmetry breaking for h = 0 (in the sense of forcing some F -or D-terms to be nonvanishing for h = 0). The most important consequence is that the contribution proportional to the top (and also bottom) Yukawa coupling is finite at one-loop and only logarithimically sensitive to the sfermion mass scale m 2 q . Indeed, since the Y T Yukawa coupling and stop soft mass terms are SU(3) symmetric the Higgs potential should be proportional to Y 2 T µ 2 T . But µ T does not lead to supersymmetry breaking for h = 0, therefore such a contribution cannot occur in STrM 4 . It can only enter the finite part of the Higgs potential such as ∆V 1−loop ∼ Y 2 T µ 2 T h 2 log(mq/f ). The mild logarithmic sensitivity to mq allows us to raise the squark masses far above 1 TeV without introducing too much fine-tuning. Logarithmically divergent contributions that arise at one loop are suppressed by electroweak gauge couplings. In our model with a low cutoff scale Λ S , these will be of the same order of magnitude as the finite part dependent on the top Yukawa couplings.
Let us first discuss the top/stop contribution to the one-loop correction ∆V 1−loop to the SM Higgs potential in our model. As usually, it is given by: where M 2 is the field dependent mass squared matrix of the theory. For a nonzero background value h of the SM Higgs field the contribution of the top sector to the fermionic mass matrix reads where we have used te abbreviations s h = sin(h/f ), c h = cos(h/f ). Hence, In the stop sector where for vanishing gauge couplings M 2 stops takes the form Let us denote the two eigenvalues of the (h dependent) top mass matrix squared (17) by t 2 1 and t 2 2 and the four eigenvalues of the stop mass squared matrix (19) by m 2 q +s 2 i , where mq is the overall scale of the soft supersymmetry breaking in the stop sector. In the following we assume 8 that m 2 q ≫ s 2 i and expand the one-loop effective potential in powers of 1/m 2 q . Up to terms of order 1/m 2 q we can rewrite the top/stop sector contribution as: where the color factor N c = 3. We also used the fact that both TrM 2 stops and TrM 2 tops do not depend on h as a consequence of SU(3) symmetry. As explained before, TrM 4 stops − 2TrM 4 tops does not depend on Y T . Therefore, in order to calculate the part of ∆V 1−loop proportional to the Yukawa couplings in the limit m 2 q ≫ s 2 i it is sufficient to find the eigenvalues of the top mass matrix squared (17).
The necessary eigenvalues of the top mass matrix are given by: The lower sign corresponds to the ordinary top quark mass, which approximately equals m t ≈ y t f sin(h/f ) where is the physical top quark Yukawa coupling. The higher eigenvalue corresponds to the mass squared of the SU(3) fermionic partner of the top quark. Inserting (22) in the last term in the curly brackets in (20) we can determine the contribution to the effective potential proportional to the top Yukawa coupling Y T . Expanding in powers of sin(h/f ) to the quartic order we find: The first term of (24) gives the Y T dependent one-loop contribution to the SM Higgs mass squared m 2 H . It is negative, which enables the electroweak symmetry breaking. Moreover, as advertised, it is not proportional to m 2 q but rather to the square of the SU(3) breaking supersymmetric parameter µ T , which may be much smaller than the soft supersymmetry breaking scale mq. The analogous contribution from the bottom-sbottom sector, can be easily obtained by using the substitutions: Terms in ∆V 1−loop logarithmically depending on the scale Λ S , i.e. proportional to STrM 4 arise from two different sources which depend on the SU(3) breaking SM gauge interactions. One is the gauge coupling dependent contribution to the sfermion mass matrices, that enters via TrM 4 stops −2TrM 4 tops in (20). In this way we get the h dependent contribution where the trace runs over all sfermions charged under U(1) Y . We have dropped the terms with fourth powers of the gauge couplings as they are small (potentially large terms ∝ g 2 2 g 2 E cancel out between the up and down type squark contributions). For non-universal soft breaking scalar masses, when Tr[Y m 2 ] = 0, this term depends quadratically on the scale mq. However it is suppressed by the small coupling g y . It is also interesting to notice that in contrast to the Little Higgs models, in which a complicated gauge structure is used to cancel the g 2 2 Λ 2 contribution to m 2 H , here it is the supersymmetric structure alone which ensures the absence of the g 2 2 m 2 q piece. The complete h dependent contribution of the gaugino/gauge boson and higgsino/Higgs boson sectors to ∆V 1−loop is rather lengthy. 9 Its most relevant parts can be approximated by We have neglected the contributions proportional to the soft mass terms of the Higgs fields and those with electroweak gauge couplings g 2 and g y in the fourth power. We have also approximated M 2 under the logarithm in (15) by µ 2 . 9 In order to compute the Higgs sector contribution to ∆V 1−loop , taking into account also heavy degrees of freedom necessary for vanishing of the h dependent part of STrM 2 , we split the neutral CP-even components of the fields in (7) and (10) into H 0 u,d +quantum fluctuations, S u,d +quantum fluctuations, compute the mass squared matrices of the quantum neutral CP-even and CP-odd and charged Higgs fields and only at the end substitute

SU (3) and electroweak breaking
We turn to analizing in a more quantitive way radiatively induced SU(3) and electroweak breaking. Our basic assumption is that the sfermion soft masses set the largest mass scale in the model. Breaking of the gauge U(1) E and the global SU(3) symmetries occurs if m 2 1 m 2 2 < m 4 3 in eq. (7). This condition is obviously satisfied for m 2 1 > 0 and m 2 2 < 0. Similarly as in the MSSM, the quantum corrections generated primarily by the top/stop loops can make the parameter m 2 Hu negative. In the present model this effect will be enhanced by the largeness of soft squark masses. The leading one-loop effect of the renormalization of m 2 Hu between the scales Λ S and mq is given by We have not displayed here the much smaller g 2 y Tr(Y m 2 ) and the gaugino contribution. Moreover, since we will assume that m 2 Hu (Λ S ) ≪ m 2 q (Λ S ), we have dropped the Higgs soft masses in the second term of eq. (27) as well. With this assumption, as long as Tr(Em 2 ) is not too large and negative, the soft mass m 2 Hu is driven towards a negative value. For very large tan β ∼ 50 one should check whether m 2 H d is not driven to negative values too but even in that case the situation can be improved for positive Tr(Em 2 ).
For tan β ≥ 5 and Y 2 T ∼ 1.3 (see below) the scale f of the SU(3) breaking can be estimated as: Since the U(1) E gauge coupling g E does not contribute to the SM Higgs mass it does not have to be small and in the following we assume g E ≈ 1. This choice minimizes the fine-tuning necesary for SU(3) breaking. Moreover, we shall soon see that increasing f increases fine-tuning in the electroweak symmetry breaking. Therefore adopt the minimal phenomenologically allowed value f ∼ 2.5 TeV [16]. Then eq. (28) sets the lower bound on the squark masses mq > 8/ log 1/2 (Λ S /mq) TeV. It also correlates the values of µ, mq and Λ S so that µ 2 ∼ 0.1m 2 q log(Λ S /mq). Essentially no fine-tuning is required for f ≈ 2.5 TeV as long as mq < ∼ 10 TeV and Λ S ∼ 100 TeV. A fine-tuning of order 1% is required for mq ∼ 50 TeV.
In the second stage we study breaking of the electroweak symmetry. The leading contributions to the SM Higgs potential are those of eq. (24), eq. (25) and eq. (26). The resulting SM Higgs doublet potential has the approximate form: At tree-level m 2 H = 0, λ = 1 8 (g 2 2 + g 2 y ) cos 2 (2β) and κ = 0. At one loop ∆V In eq. (30) the Higgs mass parameter at the electroweak scale is expressed in terms of the other parameters renormalized at the scale Λ S . In the first term we have used the tree-level relations (22) which relates the values of Y T (Λ S ) and µ T (Λ S ). We can now analyze the contribution of the first term on the rhs of eq. (30). It is quadratically dependent on the mass m T of the heavy top quark partner. For f = 2.5 TeV, taking into account the relation (31), m T has a minimum for µ T ≈ 3 TeV and Y T ≈ 1.35 at which the contribution (24) to the Higgs mass parameter is m 2 H ≈ −1.6 TeV 2 . We choose this value of µ T as it minimizes the fine-tuning in electroweak breaking. Note that these arguments do not depend on the value of stop masses mq. Also perturbativity of Y T , i.e. Y 2 T /4π < 1 up to the scale Λ S , does not impose any new bound.
Since for the electroweak breaking we need m 2 H ∼ 0.01 TeV 2 the finite Yukawa contribution is by itself too large by a factor of hundred. For our mechanism to work, it must be partly cancelled by the other contributions in eq. (30). Thus, our first conclusion is that fine-tuning at 1% level is required here. However the squark masses can be large. Note that in the MSSM for squark masses mq ∼ 10 TeV the fine-tuning is of order 0.01%.
Before discussing the remaining terms in eq. (30) it is important to note that the scale dependence of the top quark Yukawa coupling is relatively strong. Although this is formally a two-loop effect, it may introduce important corrections to our systematic one-loop calculation. To estimate this uncertainty we may use eq. (22) for y t (Λ S ) rather than y t (m t ), where y t (Λ S ) is obtained by evolving the top Yukawa coupling from the scale m t to Λ S using the appropriate RG equations. Then the corresponding numbers are µ T ≈ 3 TeV, Y T ≈ 1.15 and the contribution (24) to the mass parameter is m 2 H ≈ −1.2 TeV 2 . The fine-tuning is then slightly smaller but still of order 1%. Of course, using the present procedure for extracting y T (Λ S ) from the physical top quark mass we should, for consistency, calculate the effective potential at two loops. However, since the loop corrections to the top Yukawa coupling are likely the most important, the comparison of the two methods is at least a good estimate of the uncertainty of our results.
Turning now to the other contributions in eq. (30), the second one in importance is the µ 2 dependent part of the last term. Indeed, because eq. (28) fixes µ 2 as a function of mq and Λ S , this contribution reads approximately 10 −3 m 2 q log 2 (Λ S /mq) and for heavy sfermions it is similar in magnitude (but with opposite sign) to the Yukawa contribution. In fact, the requirement that it does not make m 2 H positive sets a stringent bound on the cutoff scale Λ S , Λ S < ∼ mqe 33 TeV/mq . For mq = 10 TeV, we get Λ S < ∼ 250 TeV. This is a postfactum justification of our assumption that the cutoff scale is not high. As for the remaining contributions in eq. (30), the term proportional to Tr[Y m 2 ] is small even for non-universal squark masses. In that case we would expect Tr[Y m 2 ] ∼ m 2 q , but it is strongly suppressed by the loop factor and the small hypercharge gauge coupling. The positive contribution of gaugino masses to m 2 H can be larger and can complement the µ 2 contribution in canceling the too large negative contribution of (24). For Λ S ≈ 100 TeV the requirement that m 2 H < 0 puts the upper bound M 2 < ∼ 4.5 TeV. Collecting all the relevant contributions to m 2 H we see that with a 1% fine-tuning the necessary value for this parameter m 2 H ∼ (100 GeV) 2 is easy to obtain for mq ∼ O(10 TeV) and Λ S ∼ O(100 TeV).
For the one-loop contributions to the quartic couplings λ and κ we get 10 From this we can estimate the SM Higgs mass: The term proportional to κ log(v 2 /2m 2 T ) turns out to be the dominant correction to the SM Higgs mass. Its effect is to change the tree-level prediction for the Higgs mass, and to raise it above M Z . Note that κ in eq. (32) is set by the top quark Yukawa coupling y t and so this contribution depends only logarithmically on m T (and thus on µ T and f ). Inserting the numbers we get the Higgs mass (for tan β ≥ 5): This is the prediction of present model. The lower value is obtained when the higher order effects associated with the RG running of the top quark Yukawa copling from m t to Λ S is taken into account. For this reason we believe it approximates slightly better the true Higgs mass in our model. There is of course also the dependence of M h on tan β, which becomes significant for tan β ≤ 5.

Conclusions
In this paper we have discussed electroweak symmetry breaking in a supersymmetric model in which the SM Higgs doublet is a pseudo-Goldstone boson of SU(3) global symmetry. The Higgs mass parameter is generated at 10 The corrections to the quartic coupling λ contained in the sin 2 (h/f ) terms in eq. (24), eq. (25) and eq. (26), once the large contribution to m 2 H proportional to Y 2 T µ 2 T is canceled against other contributions, becomes of order m 2 H /6f 2 ∼ −(100 GeV) 2 /6(2.5 TeV) 2 and is negligible. one loop level by two different, moderately fine-tuned sources of the global symmetry breaking. The mechanism works well for heavy sfermion masses mq ∼ 10 TeV, but the fine-tuning is, nevertheless, of order 1%, two orders of magnitude less than in the MSSM with similar sfermion masses. The scale Λ S at which supersymmetry breaking is mediated to the visible sector has to be low, of order 100 TeV.
Several of the phenomenological consequences of our scenario are similar to those of the so-called split supersymmetry model of Arkani-Hamed and Dimopoulos [2]. The sfermions as well as additional scalars in the Higgs sector should be beyond reach of the LHC. Also, the heavy top quark partner has mass m T > ∼ 4 TeV, too large to be seen in the LHC [20]. Chargino and neutralino masses are more model dependent but they should be smaller than 4 TeV. The important difference with the split supersymmetry scenario is that although the gluino mass is not bounded by the mechanism of the electroweak symmetry breaking, gluino is not expected to be long lived because squarks are not so heavy. The Higgs boson mass is predicted (for tan β ≫ 1) in the range 120 − 135 GeV, whereas it is in the range 130 − 170 GeV in split supersymmetry [21]. Another potential signature of our mechanism is the presence of the Z ′ gauge boson with m Z ′ ∼ 3 TeV, which should easily be discovered in the LHC [20].